A R C H I V E
VOL. LXI
O F
M E C H A N I C A L
2014
E N G I N E E R I N G
Number 4
10.2478/meceng-2014-0034
Key words: stress-strain state, pipeline, Poisson ratio, plastic deformation, anisotropy, inner pressure, strain sensor,
stress-strain diagram
JERZY MALACHOWSKI ∗ , VOLODYMYR HUTSAYLYUK ∗∗ , PETR YUKHUMETS ∗∗∗ ,
ROMAN DMITRYENKO ∗∗∗ , GRIGORII BELIAIEV ∗∗∗ , IHOR PRUDKII ∗∗∗
INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESS
PIPE IN THE INITIAL STATE
Mechanical properties of the pipeline samples that had been cut in annular and
axial directions were investigated. The methodology of modeling and calculation of
the real stress-strain state was described. The stable state during in the deformation
process was defined. The results of the experimental researches were used as a test
variant during examination of pipe strength.
1. Introduction
Pipeline systems are important for effective functioning of the industry
and as well as for providing the needs of the population with essential energy
resources: oil, natural gas, liquid petroleum, etc. An extensive network of
pipelines, both local and internationally are supporting vital functions and
economic development. Destruction of even small sections of the pipeline
could have cause serious consequences. Breakdown of pipelines is often
accompanied by explosions and fires. Total damages would result in costconcerned with a loss of gas and petroleum products, high costs of the
repair and breaking cutting the gas supply. The last one may be particularly
noticeable due to the fact that pipelines are usually located at the areas
remote from settlements, which increases the time of the repair. Moreover,
∗
Military University of Technology, Faculty of Mechanical Engineering, Department of
Mechanics and Applied Computer Science, Gen. S. Kaliskiego 2. St., 00-908 Warsaw, Poland;
E-mail: [email protected]
∗∗
Military University of Technology, Faculty of Mechanical Engineering, Department
of Machine Design, Gen. S. Kaliskiego 2. St., 00-908 Warsaw, Poland; E-mail: [email protected]
∗∗∗
The E.O.Paton Electric Welding Institute, Department of new structural forms of welded structures and designs, No.12, Bozhenko 17.St., Kiev, Ukraine; E-mail: [email protected]
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596 J. MALACHOWSKI, V. HUTSAYLYUK, P. YUKHUMETS, R. DMITRYENKO, G. BELIAIEV, I. PRUDKII
by increasing pipe diameter and operating pressure in case of an accident it
can lead to a significant environmental pollution caused as a result from a
spill of large volumes of oil or gas.
Considering life-time of the pipeline and the fact that more than 20% of
the pipes active service life will be expired shortly, the problem of investigating the causes of failure of the transport systems is becoming increasingly
important.
A failure of the bearing elements of piping systems occurs for a variety
of reasons. It depends on the design of the elements, features of their work
and loading, and physical-mechanical properties of the base material, etc.
[1]. The main pipeline remains in a complex stress state under high internal
pressure, temperature difference, uneven ground pressure, elastic distortions
that result from the rugged topography, soil impacts associated with changes
in its structure-swelling, drawdown, etc. These factors cause stresses from
bending with stretching or bending with compression in the pipeline.
One of the main criteria that determine the reliability of pipelines are
the parameters of ultimate deformation. The change in the stress-strain state
(SSS) to the level of metal plastic deformation leads to a change in geometry
of the pipe and weakens the overall strength of the construction. Problem
of investigating the real stress-strain state is that, a pipeline is under the
influence of multiaxial stresses, which change during operation time. Definition of all real values stresses for the mathematical design is not always
possible. Generally, for design of strength of the material in these cases, used
a simplified scheme [2], where the stresses are measured in two planes of
measurement.
2. Experimental and design works
The object of investigation of this article is the seamless hot pipe produced by ”INTERPIPE Nizhnedneprovsky TRUBOPRAKATNY PLANT.”
Its certificate is as follows: pipe N o 443, smelting number 32416, diameter
219 mm, wall thickness 6 mm, weight per meter ∼33.51 kg, Steel 20. A certificate of pipe quality corresponds to GOST 8732-78 and GOST 8731-74.
The mechanical properties and chemical composition are shown in Table 1
and Table 2, respectively.
Table 1.
Mechanical properties
Melt N o 32416
σ, MPa
σ0.2 , MPa
δ, %
Flattening
475.78
323.73
32.0
satisfactory
480.69
328.64
33.0
satisfactory
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INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESS PIPE IN THE INITIAL STATE
597
Table 2.
Chemical composition
Certificate(Melt
N o 32416)
Measured
(lab. EWI).
C, %
Mn, %
Si, %
S, %
P, %
Cr, %
Ni, %
Cu, %
0.19
0.54
0.29
0.02
0.011
0.07
0.05
0.08
0.177
0.55
0.289
0.018
0.008
0.078
0.065
0.070
Mechanical tensile tests were carried out on samples in the initial state
(state of delivery). At least five selected samples were cut from tube N o 443
in the annular and axial directions (Fig. 1).
Fig. 1. Schematic representation of the pipe
The first batch of samples consisted of four pieces cut in the circumferential direction. Two of them were straightened on a press, and the other
two were not rectified with a working part. Only the place under the grips
was aligned (Fig. 2a). The second batch consisted of two samples cut in the
axial direction. The surface of the samples was not treated. The tests were
conducted at the GS Pisarenko Institute for Problems of Strength of NASU.
Fig. 2. Photo (a) and schematic illustration (b) of samples cut in the circumferential direction
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598 J. MALACHOWSKI, V. HUTSAYLYUK, P. YUKHUMETS, R. DMITRYENKO, G. BELIAIEV, I. PRUDKII
The deformations in tensile specimens were defined through readings
of movements recorded by a strain sensor. The sensor was mounted on the
working parts of the samples. The strain sensor base was 25 mm. To measure
the residual deformation of the samples, the base of length of approximately
40 mm was used (Fig. 3). The cross-section area of the samples was about
103 mm2 .
Fig. 3. Photo of the strain sensor mounted on the sample
During the mechanical test of the specimen a tension curve σ̄, ε̄ was
prepared, where σ̄ – stress calculated as the quotient of force on the area
of cross-section sample in origin state, and ε̄ = ∆l/l0 regular longitudinal
deformation of the specimen, defined as base elongation divided by the length
of the base stamped on the working part. The maximum stress that was
realized was called the temporary resistance. Before this value is reached,
the specimen deforms uniformly, after that a neck is formed.
For application of the test results, not only in the case of a uniaxial stress
state, but also for the SSS, from standard curve of tension σ̄, ε̄ get real curve
of tension σr , εr (1, 2). This assertion is valid only until the moment when
neck starts to form:
εr = ln (1 + ε̄) , σr = σ̄ (1 + ε̄)
(1)
During the formation of the neck, a multiaxial stress state develops, which
depends on the geometric shape of the specimen. The next curve is transformed into the so-called real strain curve – σi , εi by applying formula (3)
[3]:
1 − 2µ
εi = εr −
σr , σi = σr
(2)
3E
where µ is the Poisson’s ratio.
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INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESS PIPE IN THE INITIAL STATE
599
In the elastic region, the Poisson’s ratio is about 0.3. Its further change
is defined by formula (3). However, the increase is observed only at the
initial stage prior to the yield strength εi0.2 . It results from generation of new
sources of dislocation and an increase in their density [7]. An example of
calculation of Poisson’s ratio µ through σ̄, ε̄, is shown in Figs 4a and 4b.
1 1 − 2µe σr
−
(3)
2
2E
εr
where µe , E − coefficient and modulus of elasticity in the range.
Residual deformation after the destruction of the sample was measured
on two different bases including a neck. This makes possible to determine
residual deformation for any base length. Outside the neck, which takes a
significant part of the deformation, the sample is deformed unevenly, and the
strain does not correspond to the tensile strength. As it is known, for the
values greater than σ̄u , the force that stretches the sample begins to fall.
With the increasing base, the residual deformation will tend to become
uniform and in infinity they will be equal mutually. Very significant deformation concentrates in the center of the neck, measurements in this place
have not been conducted.
The thin-walled cylindrical and spherical shells lose their ability to sustainable uniform deformation, where their internal pressure does not increases
(dp = 0) [8]. When shell wall local thinning takes place, it results in the loss
of stability of the original form and initial equilibrium of construction.
µ=
3. Results and Discussion
Figure 4a shows the tension curve obtained from the experimental data
and adapted her for real stresses and strain in axial and ring samples. In the
lower part figure, there is shown the calculated change in Poisson’s ratio [3].
It is worth noting that its initial value is 0.3. Curves are obtained up to start
forming neck. Figure 4b present the tension curve and the calculated value
of Poisson’s ratio up to 0.05% strain.
Points on the tensile diagram correspond to the maximum value (Fig. 4).
They present horizontal part on the real curves, respectively.
Elastic deformation (indicated by E in the figure) was obtained on the
assumption that the modulus of elasticity E = 2·105 MPa. For circumferential
and axial specimens, the proportional limit corresponds to the value of the
order of 200 MPa.
Relations rate of strain from the level of deformation is shown in Fig. 5a.
Relations of the deformation time, from the level of deformation is shown
in Fig. 5b. Tensile strain rate was defined as ∆ε̄/∆t and plotted as a function
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600 J. MALACHOWSKI, V. HUTSAYLYUK, P. YUKHUMETS, R. DMITRYENKO, G. BELIAIEV, I. PRUDKII
Fig. 4. Stress-strain diagrams σ̄, ε̄, actual diagram σi , εi in the ring and axial directions, µ –
change of Poisson’s ratio to 22% strain (a), up to 0.05% strain (b)
Fig. 5. Strain rate dependence (a) and time plastic deformation (b) from degree of deformation
ε̄, εi
of ε̄. Strain rate, defined as ∆εi /∆t and built according to εi corresponds to
real values. The curves are obtained up to strain values of 22%.
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INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESS PIPE IN THE INITIAL STATE
601
Extension rate of the working parts of the samples in the elastic region, before the formation of neck occurred, was less than 2·10−4 , 3·10−4 ,
7·10−4 s−1 for the circular sample and 2·10−4 , 6·10−4 , 4·10−4 s−1 for the axial
sample. In the initial part of the plastic deformation for the axial sample, the
yield plateau is observed.
Figure 6 shows the partial curves built for the circular and axial samples
with the strain of not greater than 5%. The number of measurements performed by the strain sensor is about 12 measurements in the first 5 seconds
of stretching and 12 measurements in the last 100 seconds. The total number
of measurements for the axial sample is 103 and 116 for the circular one.
Fig. 6. Relations of strain rate (a) and time of plastic deformation (b) from level of deformation
in the interval deformation 0–5%
The graphs show that the pipe in both circular and axial directions indicates peculiar anisotropy. If the curves obtained for the circular direction
moves to the right direction by 0.02, they approximately will be the same
with the curves obtained from the axial direction. In the circular direction,
the pipe deformed up to 2%, which is associated with its production. It is
known that after the manufacturing process, an internal pressure test was not
carried out.
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602 J. MALACHOWSKI, V. HUTSAYLYUK, P. YUKHUMETS, R. DMITRYENKO, G. BELIAIEV, I. PRUDKII
Table 3 shows the mechanical properties obtained by testing tensile specimens.
Table 3.
The mechanical properties obtained by testing tensile specimens
Ring samples
Without
Deformation
deformation
working part (I)
working part (II)
a
b
a
b∗
σ̄y , MPa
Axial samples
(III)
a∗
b
320
320
σ̄0.2 , MPa
320
327
315
308.5
σ̄u , MPa
473.21
475.5
479.34
474.76
462.39
466.67
δ, % along base of extensometer
–
38.99
35.96
41.04
–
44.79
δ, % along base of sample
26.76
33.35
33.77
33.13
40.97
37.73
δ, % uniform deformation
13.9
20.3
17.2
18.6
21.2
18.8
ψ, % relative reduction
44.52
50.79
48.98
48.24
54.6
54.9
∗
– samples for constructing curves. Base of stress sensor – 25 mm. Base of sample – 41 mm.
Table 3 shows that for both directions ”yield stress” σ̄0.2 and tensile
strengths σ̄u are the same. Figure 7 shows the stress-strain curve of six
samples.
Fig. 7. The stress-strain curve of six samples. On the descending part of the curve of samples Ia
and IIIa*, stress sensor was removed
Determination of the steady state during the deformation was carried
out according to the method of calculation for thin-walled cylindrical and
spherical shells loaded by internal pressure. For this purpose, the curve data
σi , εi for ring samples was used.
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INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESS PIPE IN THE INITIAL STATE
603
Approximation of the actual strain curve of steel (Fig. 8) σi , εi , where
σi = σ̄ (1 + ε̄) εi = ln (1 + ε̄) , from σy , εy to start of neck formation σiu , εiu
can be calculated with the use of the following equation:
σi = σ y
εi
εy
!m
(4)
where σy , εy = σ0,2 , ε0,2 , m = 0.154253.
Fig. 8. Phase stable deformation sample made from thin-walled cylindrical and spherical shells
that was loaded by internal pressure (a), scaled part curve (b)
Table 4.
Maximal values intensity of strain and stress stable deformation
σi = 740.0249εi0.154253
σi = 87.17845 ln ε i + 715.03
I
II
εi
σi , MPa
εi
σi , MPa
Stretched sample
0.1542
554.6332
0.1574
553.8401
Spherical shell
0.1028
521.0067
0.1111
523.4706
Cylindrical shell
0.0892
509.7262
0.0982
512.7107
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604 J. MALACHOWSKI, V. HUTSAYLYUK, P. YUKHUMETS, R. DMITRYENKO, G. BELIAIEV, I. PRUDKII
For thin-walled, cylindrical and spherical shells loaded by internal pressure with the initial radius equal to r0 , under large plastic deformations, the
expression for the intensity of stress σi and logarithmic strain εi are actual
for cylindrical shell:
√
!
3 Pr
2
r
, εi = √ ln
(5)
σi =
2 s
r0
3
where P − pressure; r0 and r − initiate and mean radius shell; s – original
wall thickness for spherical shell:
!
Pr
r
σi =
, εi = 2 ln
(6)
2s
r0
When the loss of stability of the initial equilibrium shape of shells occurs,
values r/r0 for cylindrical and spherical shells can be obtained from the
following expressions. [3]:
√
!
m

 3 m
r
(7)
= exp 
√ − B  = exp
r0
2
2
3
"
!#
m
r
1 2
= exp
m − B = exp
(8)
r0
2 3
3
where m = 0.154253.
The calculations show that for spherical shells relation r/r0 is smaller
than for cylindrical ones. Next, the dependence between the pressure P and
the mean radius r for cylindrical and spherical shells is obtained [3]:
– cylindrical:
!m
2 r0 s0
2
r
P = √ 2 A B + √ ln
(9)
3 r
3 r0
– spherical:
!m
2Arι̂2 s0
r
P=
B + 2 ln
(10)
r0
r3
where s0 – original wall thickness, m = 0.154253, B – constant.
It is worth pointing out that in the derivation of these calculations, relations were used obtained in the condition of incompressibility of the material
for µ = 0.5. For cylindrical shell formula: r0 s0 = rs, and for spherical shell
approximation: r02 s0 = r 2 s. Thus it is possible to obtain pressure P∗ , when
loss of stability of equilibrium initial form shells occurs. Assuming that B = 0
the following expression is obtained for cylindrical shells:
!m
2
As0
m
s0
∗
P = √
= 504.3225
(11)
√
r0
3 r0 exp (m)
3
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INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESS PIPE IN THE INITIAL STATE
605
for spherical shells:
2m
2As0
P =
r0 exp (m) 3
∗
!m
= 893.1103
s0
r0
(12)
Hence, P∗ = 28.413 MPa and P∗ = 50.31607 MPa for cylindrical and spherical shells, respectively. The spherical shell can withstand the pressure about
1.770911 times more than the cylindrical one. By using applying obtained
results, it is possible to develop a practical application method of calculation
of the mechanical properties of metal structures and criteria for their reliability. The results of experimental studies can be used as screening criteria
during the examination of structures of cylindrical shape. After production
of containers, as well as during their operation, their load span pressure Ps ,
which exceeds the working pressure Pw 1.5 times. If the container is built
out of steel, where the ratio of tensile strength to yield strength ≥ 2, the
pressure can be reduced to 1.5 times [6].
According to 1.9 [4], pipes of all types that work under pressure, must
withstand a hydraulic pressure test that is calculated according to the formula
given in [5]. Where, R – allowable stress equal to 40% of tensile strength
for this steel, s – minimal (including minus tolerance) wall thickness of the
pipe, mm and D – nominal outside diameter of the pipe, mm.
Dc = D − s; – calculated diameter of the pipe, mm.
6
s
=
= 0.027.
D 219
Taking into account the nominal outside diameter (219 mm), minimum acceptable standard value of tensile strength (412 MPa) and nominal wall thickness (6 mm) the following values are obtained:
Pt =
2sR
= 9.285 MPa
Dp
(13)
Pt
= 6.19 MPa
(14)
1.5
Taking into consideration the average value of tensile strength obtained in
the samples cut in the circumferential direction (475.7 MPa), the values of
10.72 and 7.147 MPa, are obtained for Pt and Pc , respectively.
Pc =
4. Conclusions
The article presents the stress-strain curve of the samples before the start
of necking. The identical samples were cut in the circular direction. The
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606 J. MALACHOWSKI, V. HUTSAYLYUK, P. YUKHUMETS, R. DMITRYENKO, G. BELIAIEV, I. PRUDKII
curves tension samples cut in the axial direction are characterized with a
yield plateau. The diagrams show that the pipe in both the circumferential
and axial directions indicates peculiar anisotropy. If the curve obtained for
the circular samples moves to the right direction by 0.02, they approximately
will be the same with the curves obtained from the axial direction. In the
circular direction, the pipe deformed up to 2%, which is associated with its
production.
Determination of the steady state during the deformation was carried
out according to the method of calculation for thin-walled cylindrical and
spherical shells, that was loaded by internal pressure.
Hence, P∗ = 28.413 MPa and P∗ = 50.31607 MPa for cylindrical and
spherical shells, respectively. The spherical shell can withstand the pressure
about 1.770911 times more than the cylindrical one.
The results of experimental studies can be used as screening criteria
during the examination of structures of cylindrical shape.
Taking into account the nominal outside diameter (219 mm), minimum
acceptable standard value of tensile strength (412 MPa) and nominal wall
thickness (6 mm) the following values are obtained:
Pt = 9.285 MPa;
Pc = 6.19 MPa;
Taking into consideration the average value of tensile strength obtained in the
samples cut in the circumferential direction (475.7 MPa), the values 10.72
and 7.147 MPa are obtained for Pt and Pc , respectively.
Acknowledgment
The authors gratefully acknowledge the funding from Seventh Framework Program, Marie Curie Actions: “Innovative nondestructive testing and
advanced composite repair of pipelines with volumetric surface defects”;
Acronym: INNOPIPES; Proposal Number: 318874; Grant Agreement Number: PIRSES-GA-2012-318874
Manuscript received by Editorial Board, May 14, 2014;
final version, September 28, 2014.
REFERENCES
[1] Borodavkin P.P., Edited by Sinykov A.M.: Strength of pipelines. Moskow, Nedra, 1984.
[2] Androno I.N., Kuzbogev A.S., Aginey R.V.: Resource of ground pipelines – I: Factors that
constraining resource. Standard Test Methods, Ukhta, 2008, pp. 75-96.
[3] Malinin N.N.: Applied theory of plasticity and creep. Second Edition. Moskow, Mechanical
engineering, 1975.
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INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESS PIPE IN THE INITIAL STATE
607
[4] GOST 8731-74 – Steel seamless hot rolled pipes. Technical requirements.
[5] GOST 3845-75 – Metal pipes. Method of hydraulic pressure test.
[6] Safety regulations 03-576-03 – The rules for design and safe operation of vessels that are
working under pressure.
[7] Frost H.J., Ashby M.F.: Deformation-mechanism maps. Chelyabinsk, Metallurgy, 1989.
[8] Mildin A.M., Zelentsov D.G., Olevskiy V.I., Pletin V.V.: Developing the concept of prediction the mechanical properties of thin-walled shells. Eastern-European Journal of Enterprise
Technologies, 2011, Vol. 1, No. 3, pp. 57-61.
Badanie naprężeń i odkształceń rury bezszwowej w stanie wyjściowym
Streszczenie
W pracy przedstawiono wyniki badań właściwości mechanicznych materiału z próbek wyciętych z rury bezszwowej na kierunku obwodowym i osiowym. Zaproponowano metodykę modelowania i szacowania stanu naprężenia i odkształcenia. Jest ona zgodna z metodologią stosowaną do
analiz wytrzymałościowych cienkościennych cylindrycznych i sferycznych zbiorników poddanych
ciśnieniu wewnętrznemu. W weryfikacji wyników wykorzystano wyniki badań doświadczalnych.
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